Question

Let f(x) $ = \left( {x + |x|} \right)|x|, $ then for all x?
A. f is continuous
B. f is differentiable for some x
C. f’ is continuous
D. f” is continuous

Answer 1

Hint: We will connect the given information into the form of function say f(x), for $ x \geqslant 0,x < 0, $ we will find f(x) then we will apply left hand limit and right hand side limit. Then we will find f’(x). If LHL=RHL then f(x) or f’(x) are said to be continuous otherwise they are not continuous.

Complete step-by-step answer:
If $ x \geqslant 0, $ then f(x) $ = \left( {x + x} \right)|x| $
If $ x < 0, $ then f(x) $ = x - \left( x \right)\left( { - x} \right) $
f(x) $ = 0 \times \left( { - x} \right) $
$ f(x)= 0 $
$ f(x)= 2{x^2},\,\,\,x \geqslant 0 $
 $ 0,\,\,x < 0 $
Then will take limit both side, separately,
LHL $ = $ lim f(x) $ = 0 $
RHL\[ = _{x \to {0^ + }}^{\lim }f\left( x \right)\]

RHL\[ = _{x \to {0^ + }}^{\lim }f\left( h \right)\]

RHL\[ =_{x \to {0^ + }}^{\lim }2{h^2}\]
RHL $ = 0 $
Therefore f is continuous.
Now, we will take differentiate of f(x), then
$ f’(x)= 4x $ for x>0 and
$ f’(x)= 0 $ for x<0
We will take limit both sides, we get
LHL $ = $ lim f(x)
LHL $ = 0 $
RHL\[ = _{x \to {0^ + }}^{\lim }f'\left( x \right)\]

RHL\[ = _{x \to {0^ + }}^{\lim }f\left( h \right)\]

RHL\[{ = ^{}}_{h \to {0^{}}}^{\lim }4h\]
RHL $ = 0 $
Therefor $ f’$ is continuous
Hence, the correct option is (A) and (C).
So, the correct answer is “OptionA AND C”.

Note: Students should carefully solve if the LHL and RHL derivations are not equal at any point then, the function is not differentiable at those points.